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VOLUME 74, NUMBER 8
PHYSICAL REVIEW LETTERS
20 FEBRUARY 1995
Atomic Tunneling from a Scanning-Tunneling or Atomic-Force Microscope Tip:
Dissipative Quantum Effects from Phonons
Ard A. Louis and James P. Sethna
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501
(Received 27 May 1994; revised manuscript received 18 October 1994)
We study the effects of phonons on the tunneling of an atom between two surfaces. In contrast to
an atom tunneling in the bulk, the phonons couple very strongly and qualitatively change the tunneling
behavior. This is the first example of Ohmic coupling from phonons for a two-state system. We
propose an experiment in which an atom tunnels from the tip of an scanning-tunneling microscope, and
show how its behavior would be similar to the macroscopic quantum coherence behavior predicted for
SQUIDs. The ability to tune and calculate many parameters would lead to detailed tests of the standard
theories.
PACS numbers: 61.16.Ch, 03.65.Bz, 63.20.Mt, 68.35.Wm
The fascinating experiments carried out by Eigler and
co-workers [1–4] have shown that scanning-tunneling
microscopy can be used not only for imaging, but also
to directly demonstrate fundamental aspects of quantum
mechanics. Inspired by their success, we propose an experiment to test the effect of a dissipative environment on a
quantum mechanical system. While often ignored in many
applications of quantum mechanics, the environment can
have important effects on a simple quantum system. In
fact, from the dawn of quantum mechanics, people have
been attributing the collapse of the wave function to the
interaction of a quantum system with a macroscopic environment [5]. As recently emphasized by Leggett [6],
condensed-matter physics provides a natural laboratory
for studying this: How does the surrounding macroscopic
crystalline environment change the behavior of an embedded quantum system? In particular, what can realistic
models of the environment tell us?
A tantalizing view of the importance of the environment is given by the “overlap catastrophe” [7]. In a metal,
the conduction electrons must be rearranged when the
quantum subsystem changes state: The initial and final
electron ground-state wave functions are so different that
the bath of conduction electrons can force the subsystem
to dissipate energy during each transition, or even keep
the transition from occurring. In this Letter, we show that
phonons, an environment even more ubiquitous than conduction electrons, can also have an overlap catastrophe.
In particular, we show that quantum tunneling behavior
between two surfaces can be dramatically altered by the
coupling to phonons. We show how detailed calculations
of the macroscopic quantum coherence (MQC) community [8] could be tested experimentally in an admittedly
microscopic system using the tunneling of atoms onto and
off of the tip of a scanning-tunneling microscope (STM)
or atomic-force microscope (AFM).
Eigler and co-workers report on an “atomic switch,” in
which a Xe atom is reversibly transferred from a Ni(110)
surface to the W tip, as well as on “transfer near contact”
0031-9007y 95 y 74(8) y 1363(4)$06.00
processes in which an STM tip is brought so close to a
surface that the absorbed Xe atom spontaneously hops to
it [2,3,9]. Several theories have been put forth to explain
the atomic switch experiment [10–14]. The potentials
calculated for the Xe-tip-surface system have a doublewell shape, as depicted in Fig. 1, and bringing the tip just
slightly closer to the surface than the switch operating
conditions can give significant tunneling amplitudes, even
for a heavy atom such as Xe. If we take, for example, some
calculated parameters from Ref. [14] for Xe on Ni(110)
sV0 ­ 7 MeV, Q0 ­ 0.6 Åd, we can obtain tunneling elements up to h̄DykB , 0.5 K. Here we use the example
of Xe simply because a lot is known about that system.
Lighter elements such as He would have even larger
tunneling amplitudes. By adding an electric field, the wells
can be biased in either direction quite sensitively, as was
shown in Ref. [11], so that by varying the tip position
and the bias both D and e can be varied independently.
Of course the exponential dependence of the tunneling
FIG. 1. Double-well potential for the tunneling atom. The
atom tunnels from the STM or AFM tip to the substrate
surface and back under the influence of a potential such as the
one shown above. For temperatures kB T ø h̄v0 y2 , V0 , the
tunneling is between the lowest harmonic oscillator type states
in each well. The tunneling amplitude h̄D gives the energy
splitting of the symmetric and antisymmetric superpositions of
tip 1 surface states, and is determined by the barrier height V0 ,
the effective distance Q0 , the mass of the particle, as well as the
possible asymmetry energy e . The small oscillation frequency
v0 is determined by the same parameters.
© 1995 The American Physical Society
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VOLUME 74, NUMBER 8
PHYSICAL REVIEW LETTERS
amplitude h̄D on system parameters and the precise shape
of the potential makes it very difficult to accurately predict
its magnitude. The point is more so because the atom can
be brought arbitrarily close to where it sticks to the surface,
the tunneling rate can be made arbitrarily large.
We now proceed to the main part of this Letter:
the effect of the phonon environment on our quantum
tunneling system, and consider temperatures such that
kB T ø h̄v0 y2 , V0 , allowing us to truncate the Hilbert
space to two states, one for each well [15].
As depicted in Fig. 2, the (AFM or STM) 1 atom
system will exert a different force 6DFy2 on the surface
depending on where the atom is. The phonons will relax,
switching their equilibrium positions in response to the
atom being on the tip or the surface. This can be modeled
by the Hamiltonian of the two-state system with the
displaced phonons
h̄D
e
s x 1 sz
H ­
2
2
∑
∏
X 1
1
2
2
mks xks
1
1 1 mks vks
sxks 1 1 qks sx d2 .
2
ks 2
(1)
The hxks j are the normal coordinates for a given polarization s and wave vector k. The masses of the normal
coordinate particles are denoted by mks , and the phonon
angular frequency is vks . The hqks j are the new equilibrium positions in the presence of the force 6DFy2.
In the absence of phonons, the probability of finding
the atom at the tip at time t , given that it was started
there at time t ­ 0, oscillates back and forth with the
usual form Pstd ­ sin2 sDtd. What now is the effect
of the phonons (environment) on the behavior of the
atom (embedded quantum system)? Is there a qualitative
change of behavior? For atoms tunneling between two
states in the bulk, an approach often taken to account
for the environment is calculating the overlap integral of
the atom in state one 1 relaxed phonons with the atom
in state two 1 relaxed phonons. The new renormalized
FIG. 2. Schematic drawing of the (STM or AFM) 1 atom
system. The tunneling of the atom couples to phonons via
the response of the substrate to the external force 6DFy2
that depends on the position of the atom. The coupling to
an environment drastically alters the tunneling behavior. For
small coupling and low temperature, the tunneling is coherent
with a renormalized tunneling frequency. For larger coupling
and zero temperature, there is no tunneling at all, while at
finite temperature there is incoherent tunneling with a power
law dependence on temperature.
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20 FEBRUARY 1995
tunneling amplitude is just the bare amplitude multiplied
by a Frank-Condon phonon-overlap factor. The atom
still oscillates back and forth, but now with a reduced
rate. For atoms tunneling in the bulk, this approach gives
good qualitative results [16,17]. However, for an atom
tunneling between surfaces, the situation is quite different,
as can be seen by considering the force exerted by the
atom on its environment. While the defect tunneling in
a solid only exerts a dipole force, resulting in qk , k 21
[16], the external tip 1 atom system exerts a monopole
force on the surface, resulting in qk , k 22 [18,19]. The
external case has therefore a stronger coupling at low
frequencies, and a naive calculation of the Frank-Condon
factor gives an infrared divergence, a tell-tale sign that the
adiabatic approximation breaks down. In fact, the case
of tunneling between surfaces corresponds to “Ohmic”
dissipation in the language of the macroscopic quantum
tunneling (MQT) literature, as opposed to the bulk case,
where the dissipation is of the “super-Ohmic” variety [8].
The effect of the environment can now be characterized
by a dimensionless coupling parameter a defined as
hQ02
,
a­
(2)
2p h̄
with h equivalent to the friction coefficient in the macroscopic limit [20]. The tunneling element is renormalized
to Dr ­ DsDyvc days12ad for a , 1, and is zero for a . 1
[21]. In other words, for small coupling parameter a ,
the effect of the phonons is quantitative only, reducing
the tunneling frequency, while if a crosses 1, the effect
is qualitative: there is a transition to no tunneling at all.
This fascinating effect is the phonon analog of Anderson’s
overlap catastrophe in an electron gas [7]. We emphasize
that this would be the first case in which an overlap catastrophe is caused by phonons [22].
Having shown that an atom tunneling from an STM or
AFM tip to a surface undergoes Ohmic dissipation from
the phonon environment, we come to the question: Is
the dimensionless coupling parameter a in an interesting
regime? Assuming a point force DFy2 on a semi-infinite
isotropic medium with a linear (Debye) dispersion we find
1
a­
Gssd sDFd2,
(3)
2
64p h̄rct3
where r is the density in kg m23 , ct is the transverse
sound velocity, and Gssd is a tabulated function of
Poisson’s ratio s ­ s3l 1 mdf6sl 1 mdg, with roughly
equal contributions from the acoustic bulk and Rayleigh
modes [23]. The critical DF for which a ­ 1 goes from
about 3 nN for W to 0.3 nN for Pb. Another order of
magnitude reduction can be obtained for some organic
materials. At the short distances we need for atomic
tunneling, the typical force of a tip on a surface is of the
order of a nN per Å separation [24], putting us right into
the interesting regime [25].
We envisage a number of realizations of the atomsurface tunneling experiments. First, it has been calculated that for kB T $ h̄Dypa the coherent oscillations are
VOLUME 74, NUMBER 8
PHYSICAL REVIEW LETTERS
destroyed [26,27], and there is only incoherent tunneling
with a rate [8]
µ
∂
Gsad
spd2a11y2 kB T 2a21
t 21 ­ Dr
, (4)
Gsa 1 1y2d
2
h̄Dr
from which the parameter a could be extracted and compared to our predictions. (For example, an AFM operating in the noncontact mode can measure the oscillations
by the force modulation of 6DFy2. The tunneling current of an STM is significantly larger when the atom is on
the tip rather than when it is on the surface [2].) Beautiful
experiments [28] on individual two-state tunneling defects
coupled to conduction electrons in mesoscopic metals and
on SQUIDs [29] have confirmed this power law behavior of the rate with temperature. However, in contrast to
our system, these experiments do not allow one to vary
a , or to calculate it from first principles. We believe this
would be the first quantitative test of the a parameter,
giving valuable insight into the validity of the linear coupling model of a dissipative environment. By varying the
distance between the tip and the surface, or by varying
the position within the surface unit cell, the force DF, and
thus the coupling parameter a , could be tuned. Of course,
simultaneously, this would heavily affect the tunneling
frequency D by changing the potential barrier. However,
one could still see the qualitative change in temperature
dependence of the rate. Of particular interest would be
the crossover from decreasing with increasing temperature to increasing with increasing temperature of the rate
when a crosses 1y2 from below [30].
Second, a more ambitious experiment might test the
coherence predictions: watching the tunneling turn off as
a is increased. Coherent oscillations, one of the goals of
the MQC community [8], will occur if T , h̄DykB pa ,
and a , 1y2 [31]. There has been extensive discussion
of the problems of measuring coherent oscillations in
the MQC literature [32]. For example, Peres [33] has
claimed that noninvasive measurements are impossible
in this case: a measurement introduces an energy of
order h̄v0 . This has been disputed by Leggett and
Garg [34], and, in fact, we find that the uncertainty in
momentum P caused by measuring the position to better
than Q0 y2 is sDPd2 y2 $ h̄v0 sh̄v0 y64V0 d, which gives a
sizable window for the direct measurement we propose.
This expression can be easily derived from the positionmomentum uncertainty principle. Peres makes too crude
an approximation for DP , and thus overestimates the
effect of localizing the object to be measured (flux in his
case) on its conjugate variable [19]. More specifically,
for the case of Xe on Ni(110) theories for the switch
experiment estimate that for a bias voltage greater than
h̄v0 , about one in 2000 electrons will inelastically scatter
and excite the atom into a higher vibrational state [11].
For a current of ,0.1 nA the atom would be excited on
average about once every ms. Thus an STM in imaging
mode can only measure oscillations if 1yD ø 1 ms, a
rate that is easily reached for close enough tip-surface
20 FEBRUARY 1995
separation. If the bias is less than h̄v0 , excitations can
only occur through multiple electron transitions, so the
excitation rate will be much smaller and lower oscillation
rates can be measured.
For experimentally accessible temperatures (say 1 K)
with a ­ 0.1, this means a tunneling rate of at least
10 GHz. Recent developments in high-speed STM have
achieved a time resolution of picoseconds [35], and we
envisage a direct measurement of the correlations by
sending in pairs of voltage pulses [32]. This could
be accomplished by a low-temperature STM coupled to
picosecond lasers that generate the voltage pulses (as
was done in Ref. [35]). Varying the intrapair time by
py2D will give an oscillation of the integrated current
that reaches a maximum of 40% of the current difference
between the atom on the tip and the surface for a slightly
biased well with e ­ D [19]. The interpair time must be
much greater than the intrapair time, and the pulses must
be long compared to the electron tunneling time (which is
on the order of fs). These would be the first measurements
of the coherent oscillations of a single entity with Ohmic
dissipation, and, while the atom is not “macroscopic,” our
proposed experiment should be a fertile testing ground
of new measurement schemes, and may shed light on
the even more difficult problem of observing MQC in
SQUIDs [36].
In summary, we have shown how the phonons couple
in a fundamentally different way to particles at a surface
than to particles in a bulk, and can cause an overlap catastrophe, just as electrons can. More precisely, the coupling
of an atom to phonons at the surface produces Ohmic dissipation, and qualitatively changes the tunneling behavior,
in contrast to the well-known case of tunneling in the bulk
where the effect of phonons is only quantitative. We have
proposed an experiment with an atom tunneling between
an STM or AFM tip and a surface, and show how this
could test theories of MQC, albeit in a microscopic setting. Because the parameters D, e , and a can be varied
by changing the tip-surface position and the bias, many
different regimes of the theories of two-state systems with
Ohmic dissipation can be put to the test.
We thank Thomás Arias, Sue Coppersmith, Mike
Crommie, Shiwu Gao, and Jörg Drãger for encouragement
and discussions; Mark Stiles for pointing out to us the
importance of the Rayleigh phonons in our calculation of
a ; and Mark Stiles and Risto Nieminen for explaining
phonon coupling in atomic scattering to J.P.S. This work
was supported by NSF under Grant No. DMR-19-18065
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524 (1990).
[2] D. M. Eigler, C. P. Lutz, and W. E. Rudge, Nature (London) 352, 600 (1991).
[3] J. A. Stroscio and D. M. Eigler, Science 254, 1319 (1991).
[4] M. F. Crommie, C. P. Lutz, and D. M. Eigler, Nature
(London) 363, 524 (1993); Science 262, 218 (1993).
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PHYSICAL REVIEW LETTERS
[5] H. D. Zeh, in Foundations of Quantum Mechanics, edited
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[7] P. W. Anderson, Phys. Rev. Lett. 18, 1049 (1967).
[8] See A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987), and
references therein.
[9] Other workers have also succeeded in transferring, for
example, Si atoms reversibly to a Si surface [I.-W. Lyo
and P. Avouris, Science 253, 173 (1991)].
[10] S. Gao, M. Perrson, and B. I. Lundqvist, Solid State
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[16] J. P. Sethna, Phys. Rev. B 24, 698 (1981); 25, 5050 (1982).
[17] A similar effect is at work in the small-polaron problem,
where the phonon-overlap integral slows down the motion
of an intinerant electron.
[18] This can
by the following argument: qki ,
P be seen2ik?r
l [19], where Dskd
DF
sr
de
Dskd21
j
l
ij is the dynamil
ij
cal matrix which goes as ,k 2 for small k. Thus qk , k 21
for a dipole force, and qk , k 22 for a monopole force.
[19] A. A. Louis and J. P. Sethna (unpublished).
[20] A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149,
374 (1984); 153, 445(E) (1984).
[21] S. Chakravarty, Phys. Rev. Lett. 49, 681 (1982); A. J. Bray
and M. A. Moore, Phys. Rev. Lett. 49, 1546 (1982).
[22] Ohmic coupling from one-dimensional phonons has been
shown for an escape problem by L. S. Levitov, A. V. Shytov, and A. Yu. Yakovets, Report No. Cond-Maty9406117,
1994.
[23] Gssd varies from about 6.36 for s ­ 0 to 2.35 for
s ­ 0.5 [19]. For a different symmetry (say cubic),
the calculation becomes very difficult to do analytically.
One must resort to numerical procedures such as slab
calculations of the phonon dispersion and density of states
[see, for example, R. E. Allen, G. P. Alldredge, and F. W.
de Wette, Phys. Rev. B 4, 1661 (1971)]. However, we
do not expect the value of a to change very much
from the isotropic result as it depends only on the longwavelength elastic behavior (which is independent of any
local details) and DF, which can be measured with an
AFM.
[24] U. Durig et al., Phys. Rev. Lett. 57, 2403 (1986); 65,
349 (1990); C. J. Chen Introduction to Scanning Tunneling
Microscopy, (Oxford University Press, New York, 1993).
[25] Note that we have ignored the phonons in the tip. On
1366
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
20 FEBRUARY 1995
the one hand, if we approximate the tip as another flat
surface, the total coupling will only be the sum of the a ’s
from each surface. As the tip is usually made of tungsten
or some other stiff material, it will typically have an a
order of magnitude smaller than the a from the substrate
surface. On the other hand, if we approximate the tip as
a 1 2 d vertical line of atoms, the coupling will be subOhmic, and the tunneling will be quenched at all coupling
strengths. We have also ignored possible coupling to
electron-hole pairs. Of course for an insulator they will
not matter, and, for example, in the case of Xe on Ni(110),
the electron-hole pair dissipation is typically a factor 100
smaller than the phonon dissipation [10]. When it is
not negligible, it will give an additive contribution to a .
An estimate of the force difference DF can be obtained
from ab initio total energy pseudopotential calculations
quite accurately with standard techniques [Thomás Arias
(private communication)].
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S. Han, J. Lapointe, and J. E. Lukens, Phys. Rev. Lett. 66,
810 (1991).
We note in passing that adding an oscillating electric field in the incoherent regime would bring us into
the very interesting regime of quantum stochastic resonance, recently discovered by Löfsted and Coppersmith
[R. Löfsted and S. N. Coppersmith, Phys. Rev. Lett. 72,
1947 (1994); AT&T Bell Labs Report, 1994], as well as
a host of other effects relating to driven quantum systems
with dissipation.
Although the transition to no tunneling only occurs for
coupling a . 1, the tunneling in the region 1y2 # a , 1
is expected to be incoherent [8].
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Coherent oscillations with Ohmic coupling have been
observed by H. Wipf et al. [Europhys. Lett. 4, 1379
(1987)] for hydrogen tunneling in niobium. In this case,
the measurement is in frequency space on a large number
of trapped H atoms interacting with conduction electrons.
Recent calculations (D. S. Fisher and A. L. Moustakas,
Report No. Cond-Maty9408013, 1994) have shown that
the conduction electron bath that gives Ohmic dissipation
in the aforementioned experiments [28] will not lead to
localization due to simultaneous tunneling of the atom
and electrons, and thus a phonon bath may be the only
experimentally accessible route to this effect.